An Approached Solution of Wave Equation with Cubic Damping by Homotopy Perturbation Method (HPM), Regular Pertubation Method (RPM) and Adomian Decomposition Method (ADM)

In this study, we consider the wave equation with cubic damping with its initial conditions. Homotopy Perturbation Method (HPM), Regular Pertubation Method (RPM) and Adomian decomposition Method (ADM) are applied to this equation. Then, the solution yielding the given initial conditions is gained. Finally, the solutions obtained by each method are compared.

Linear and nonlinear phenomena are of fundamental importance in various fields of science and engineering. Most models of real-life problems, however, are still very difficult to solve. Therefore, approximate analytical solutions such as HPM, RPM and ADM were introduced, which are effective and convenient for both linear and nonlinear equations.
In this paper, we consider the following nonlinearly damped wave equation Where ε is perturbation parameter which , u(t, x) is some physical quantity, x the space variable and t stands for time.
These types of equations are of considerable significance in various fields of applied sciences, mathematical physics, nonlinear hydrodynamics, engineering physics, biophysics, human movement sciences, astrophysics and plasma physics. The paper is organised as follows : in section 1, we start with the solving (1) by HPM. In Section 2 and section 3, we construct the solution of (1) respectively by RPM (He, J. H., 2004) and ADM. Section 4 contains the comparison of the solutions obtained by the different methods.

Basic Idea of the Homotopy Perturbation Method
To illustrate the basic idea of HPM, consider the following nonlinear differential equation with boundary conditions B ( u, ∂u ∂n where A is a general differential operator, B is a boundary operator, f (r) is a known analytic function, and Γ is the boundary of the domain Ω. Generally speaking, the operator A can be decomposed into two parts L and N, where L is a linear and N is a nonlinear operator. Equation (2), therefore, can be rewritten as follows : We construct a homotopy v (r, p) : where u 0 is an initial approximation to the solution of Equation (2). In this method, we use the homotopy parameter p to expand v as a power series The approximate solution can be obtained by setting p = 1, The convergence of the series of (4) has been proved in (He, J. H., 1999;He, J. H., 2000).

Application of HPM Wave Equation with Cubic Damping
According to the HPM (He, J. H., 1999;He, J. H., 2003;He, J. H., 2004;Gupta, S. & et al., 2013), we can construct the homotopie H(v, p) for equation (1) which satisfies : Let as choose the initial approximation as u 0 = cos x, thus ∂ 2 u 0 ∂t 2 = ∂u 0 ∂t = 0 We have, Assume the solution of (1) to be in the form : Substituting (6) into (5) and equating the coefficients of like powers p, we get the following set of differential equations : p 3 : From the above equations, we can obtain In principle, it is possible to calculate more components in the expansion series to enhance the approximation. Therefore, we get the tenth-order approximation, This solution can be written in the form : − 35353 t 23 4612874112000

RPM Description
In order to show, the basic idea of RPM, consider the following differential equation where L ε is a genaral differential operator, B ε is a boundary operator, and ∂Ω In general, the equations (12)-(13) contain a very small parameter ε. In this method, we use the parameter u ε as a power series , Substituting (14) in (12)-(13), and collecting the coefficient of like powers of ε yields and equating the coefficient of each power of ε to zero. We obtain systems of recurrent boundary problems, easy to solve.
The approximate solution is given by

Application of RPM
Let us suppose that the solution u(t, x) of the initial value problem (1) has the following form (JAGER, DE. E. M., & JIANG, FU RU, 1996) : Putting (16) into (1), and collecting equal powers of ε we obtain a system of recurrent initial value problems ε 0 : . . .

Generalities
General properties of ADM and its application can be found in (ABBAOUI, K., 1995;ABBAOUI, K., & CHERRUAULT, Y., 1994;ABBAOUI, K., & CHERRUAULT, Y., 1999;NGARHASTA, N. & et al., 2002). Some of these are outlined as follows. Suppose that we need to solve the following equation in a real Hilbert space H, where A : H → H is a linear or a nonlinear operator, f ∈ H are given; and u ∈ H is the unknown. The principle of the ADM is based on the decomposition of the nonlinear operator A in the following form: where L + R is linear, N nonlinear, L invertible with L −1 as inverse. Using that decomposition, equation (21) is equivalent to where θ satisfies Lθ = 0. Equation (22) , 1999;NGARHASTA, N. & et al., 2002): where λ is a parameter used by "convenience ". Thus (22) can be rewritten as follows : We suppose that the series +∞ ∑ n=0 u n and +∞ ∑ n=0 A n are convergent, and obtained by identification the Adomian algorithm : In practice it is often difficult to calculate all the terms of an Adomian series; so we approach the series solution by the truncated series u = n ∑ i=0 u i , where the choice of n depends on error requirements.
Finally, the approximate solution of (1) is given by : This solution can be written in the form : l 1 (t) cos 3 x + l 2 (t) cos x sin 2 x ] +ε 2 [ l 3 (t) cos 5 x + l 4 (t) cos 3 x sin 2 x + l 5 (t) cos x sin 4 x ] +ε 3 [ l 6 (t) cos 7 x + l 7 (t) cos 5 x sin 2 x + l 8 (t) cos 3 x sin 4 x + l 9 (t) cos x sin 6 x ] The tables (2), (4) and (6) show absolute error between the various approximate solutions. It is noticed that the absolute error between the solutions of RPM and ADM just as RPM and HPM increases when ε becomes increasingly large.   The figures (1) and (2) give the comparison of the approximate solutions in dimension 2, obtained by the three methods.